Given $ \overrightarrow{OL}\perp\overrightarrow{ON}$, $ m \angle LOM = 9x + 32$, and $ m \angle MON = 9x + 4$, find $m\angle LOM$. $O$ $L$ $N$ $M$
Solution: From the diagram, we see that together ${\angle LOM}$ and ${\angle MON}$ form ${\angle LON}$ , so $ {m\angle LOM} + {m\angle MON} = {m\angle LON}$ Since we are given that $\overrightarrow{OL}\perp\overrightarrow{ON}$ , we know ${m\angle LON = 90}$ Substitute in the expressions that were given for each measure: $ {9x + 32} + {9x + 4} = {90}$ Combine like terms: $ 18x + 36 = 90$ Subtract $36$ from both sides: $ 18x = 54$ Divide both sides by $18$ to find $x$ $ x = 3$ Substitute $3$ for $x$ in the expression that was given for $m\angle LOM$ $ m\angle LOM = 9({3}) + 32$ Simplify: $ {m\angle LOM = 27 + 32}$ So ${m\angle LOM = 59}$.